🔶Intro to Abstract Math Unit 5 – Functions and Mappings

What Are Functions?

• Functions are mathematical objects that assign a unique output to each input in their domain
• Consist of a set of ordered pairs where no two pairs have the same first element (input)
• Can be represented using equations, graphs, or tables
• Denoted using function notation $f(x)$ where $f$ is the function name and $x$ is the input variable
• Functions map elements from one set (domain) to another set (codomain)
• The set of all outputs produced by a function is called its range, which is a subset of the codomain
• Functions are fundamental building blocks in mathematics and have wide-ranging applications in science, engineering, and economics

Types of Functions

• Linear functions have the form $f(x) = mx + b$ where $m$ is the slope and $b$ is the y-intercept (e.g., $f(x) = 2x + 1$)
• Quadratic functions have the form $f(x) = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants and $a \neq 0$ (e.g., $f(x) = x^2 - 4x + 3$)
  • The graph of a quadratic function is a parabola
• Exponential functions have the form $f(x) = a^x$ where $a$ is a positive constant not equal to 1 (e.g., $f(x) = 2^x$)
• Logarithmic functions have the form $f(x) = \log_a(x)$ where $a$ is a positive constant not equal to 1 and $x > 0$ (e.g., $f(x) = \log_2(x)$)
  • Logarithmic functions are the inverses of exponential functions
• Trigonometric functions include sine, cosine, and tangent, which relate angles to ratios of side lengths in a right triangle (e.g., $f(x) = \sin(x)$)
• Piecewise functions are defined by different equations over different intervals of the domain (e.g., $f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases}$)
• Constant functions have the same output value for every input (e.g., $f(x) = 3$)

Key Properties of Functions

• Injective (one-to-one) functions map distinct inputs to distinct outputs
  • For every element in the codomain, there is at most one element in the domain that maps to it
• Surjective (onto) functions map the domain onto the entire codomain
  • For every element in the codomain, there is at least one element in the domain that maps to it
• Bijective functions are both injective and surjective
  • There is a one-to-one correspondence between the domain and codomain
• Even functions are symmetric about the y-axis, satisfying $f(-x) = f(x)$ for all $x$ in the domain (e.g., $f(x) = x^2$)
• Odd functions are symmetric about the origin, satisfying $f(-x) = -f(x)$ for all $x$ in the domain (e.g., $f(x) = x^3$)
• Periodic functions repeat their values at regular intervals, satisfying $f(x + p) = f(x)$ for all $x$ in the domain, where $p$ is the period (e.g., $f(x) = \sin(x)$ with period $2\pi$)
• Monotonic functions are either increasing or decreasing over their entire domain
  • For increasing functions, $x_1 < x_2$ implies $f(x_1) \leq f(x_2)$ for all $x_1, x_2$ in the domain
  • For decreasing functions, $x_1 < x_2$ implies $f(x_1) \geq f(x_2)$ for all $x_1, x_2$ in the domain

Mappings and Their Significance

• Mappings are a generalization of functions that assign elements from one set (domain) to elements in another set (codomain)
• Mappings can be represented using arrow diagrams or ordered pairs
• Mappings are essential for understanding the relationships between sets and the transformations of elements from one set to another
• Mappings can be used to model various real-world phenomena, such as the assignment of tasks to employees or the transformation of inputs to outputs in a system
• The study of mappings leads to important concepts in abstract algebra, such as homomorphisms and isomorphisms
• Bijective mappings, also called one-to-one correspondences, are of particular significance as they allow for the comparison and identification of equivalent structures
• Mappings can be composed, allowing for the study of complex transformations as a sequence of simpler mappings

Composition and Inverse Functions

• Function composition combines two or more functions to create a new function
• The composition of functions $f$ and $g$ is denoted as $(f \circ g)(x) = f(g(x))$, where the output of $g$ becomes the input of $f$
• Function composition is associative: $(f \circ g) \circ h = f \circ (g \circ h)$
• Function composition is not always commutative: $f \circ g$ may not equal $g \circ f$
• The inverse of a function $f$, denoted as $f^{-1}$, "undoes" the original function: $f(f^{-1}(x)) = f^{-1}(f(x)) = x$
  • For a function to have an inverse, it must be bijective (one-to-one and onto)
• The graph of an inverse function is the reflection of the original function across the line $y = x$
• The inverse of a composition of functions is the composition of their inverses in reverse order: $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$

Domain and Codomain

• The domain of a function is the set of all possible input values
• The codomain of a function is the set of all possible output values
• The domain and codomain are essential for understanding the behavior and properties of a function
• Functions can be classified based on their domain and codomain (e.g., real-valued functions, complex-valued functions)
• Restricting the domain of a function can change its properties and behavior
  • For example, the square root function $f(x) = \sqrt{x}$ is only defined for non-negative real numbers
• The range of a function is the set of all actual output values, which is a subset of the codomain
• Determining the domain and range of a function is crucial for solving equations and inequalities involving the function

Practical Applications

• Functions are used in various fields to model and analyze real-world phenomena
• In physics, functions describe the relationships between physical quantities (e.g., position, velocity, and acceleration)
• In economics, functions model the relationships between economic variables (e.g., supply and demand curves)
• In computer science, functions are used to encapsulate reusable code and perform specific tasks
• In data analysis, functions are used to fit models to data and make predictions
• In engineering, functions describe the behavior of systems and components (e.g., transfer functions in control systems)
• Optimization problems often involve finding the maximum or minimum values of a function subject to certain constraints
• Probability density functions and cumulative distribution functions are used to describe the likelihood of events in probability theory and statistics

Common Pitfalls and How to Avoid Them

• Confusing the domain and codomain of a function
  • Always clearly specify the domain and codomain when defining a function
• Attempting to evaluate a function outside its domain
  • Check the domain of a function before evaluating it for a given input
• Misinterpreting the graph of a function
  • Pay attention to the scales and units of the axes, and consider the context of the problem
• Incorrectly applying function composition
  • Remember that the output of the inner function becomes the input of the outer function
• Forgetting to check for the existence of an inverse function
  • Verify that a function is bijective before attempting to find its inverse
• Misusing function notation
  • Use parentheses to denote function evaluation (e.g., $f(x)$) and avoid ambiguous or inconsistent notation
• Overlooking the importance of domain and range when solving equations and inequalities
  • Consider the domain and range of functions involved in the equation or inequality to avoid extraneous solutions
• Neglecting the context and units of a problem when interpreting the results
  • Always interpret the results of a function in the context of the original problem and ensure that the units are consistent